Subordination theorems for some classes of starlike fumctions

Roger W. Barnard and John Lawson Lewis

Vol. 56 (1975), No. 2, 333–366

DOI: 10.2140/pjm.1975.56.333

Abstract

Let K_r = {z : |z| < r},r > 0. For given α,0 < α < ∞,d,0 ≦ d < 1, and M,1 < M ≦∞, let S(α,d,M) denote the class of univalent and normalized α starlike functions f in K₁ with K_d ⊂ f(K₁) ⊂ K_M. The authors show the existence of a function F ∈ S(α,d,M) with the properties: (a) log F(z)∕z,z ∈ K₁, is univalent, (b) if f ∈ S(α,d,M), then log f(z)lz,z ∈ K₁, is subordinate to log F(z)lz,z ∈ K₁. Letting α → 0 they obtain a similar subordination result for normalized starlike univalent functions. They then point out that these subordination results solve and give uniqueness for a number of extremal problem in the above classes.

Mathematical Subject Classification

Primary: 30A32

Milestones

Received: 4 December 1973

Published: 1 February 1975